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<title>Surface Evolver geometric elements</title>
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<h1 class="center">
<a href="http://www.susqu.edu/brakke/evolver/evolver.htm" class="comic">
Surface Evolver</a> Documentation</h1>

<a href="evolver.htm#doc-top">Back to top of Surface Evolver documentation.</a>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<a href="index.htm">Index.</a>


<a   id="attributes"></a>
<a   id="geometric-elements"></a><h1>Geometric elements</h1>

      The surface is defined in terms of its geometric elements of 
      each dimension.  Each element has its own set of attributes.
      Some may be set by the user; others are set internally but
      may be queried by the user.  It is also possible to dynamically
      define <a href="#extra-attributes">extra attributes</a>
       for any type of element, which may
      be single values or vectors of values.  Attribute values can
be specified in the <a href="datafile.htm">datafile</a>, and queried
with <a href="syntax.htm#attr-values">commands</a>.
<p>Elements:
<ul> 
<li><a href="#vertices">Vertices</a>
<li><a href="#edges">Edges</a>
<li><a href="#facets">Facets</a>
<li><a href="#bodies">Bodies</a>
<li><a href="#facetedges">Facetedges</a>
</ul>
<hr>
<a   id="vertices"></a><h2>Vertices</h2>

            A vertex is a point in space.
            The coordinates of the vertices are the parameters
            that determine the location of the surface.  It is
            the coordinates that are changed when the surface 
            evolves.  A vertex carries no default energy, but may
            have energy by being on a 
            <a href="constrnt.htm#level-set-constraints">
            level set constraint</a> in the 
            <a href="model.htm#string-model">string model</a>,
            or by having a <a href="quants.htm">named quantity</a>
            energy applied to it.
            The vertices of the original surface
            are defined in the <a href="datafile.htm#vertices-section">
            vertices section</a> of the datafile.<p>

            Vertex attributes (grouped somewhat relatedly):
<table><tr><td style="vertical-align:text-top">
<ul>
<li>            <a href="#id">id</a>
<li>            <a href="#original">original</a>
<li>            <a href="#coordinates">coordinates, x,y,z, __x[]</a>
<li>            <a href="#vertex-edges">vertex edges</a>
<li>            <a href="#vertex-facets">vertex facets</a>
<li>            <a href="#valence,-vertex">valence</a>
<li>            <a href="#bare,-vertex">bare</a>
<li>            <a href="#fixed,-vertex">fixed</a>
<li>            <a href="#constraints,-vertex">constraints</a>
<li>            <a href="#on_constraint">on_constraint</a>
<li>            <a href="#hit_constraint">hit_constraint</a>
<li>            <a href="#value_of_constraint">value_of_constraint</a>
<li>            <a href="#v_constraint_list">v_constraint_list</a>
<li>            <a href="#constraint-normal">constraint normal</a>
<li>            <a href="#boundary,-vertex">boundary</a>
<li>            <a href="#parameter-values">parameter values, p1,p2, p[]</a>
<li>            <a href="#on_boundary">on_boundary</a>
<li>            <a href="#v_boundary">v_boundary</a>
</ul></td><td style="vertical-align:text-top"><ul>
<li>            <a href="#extra-attributes">extra attributes</a>
<li>            <a href="#quantity-attribute">named quantities</a>
<li>            <a href="#on_quantity">on_quantity</a>
<li>            <a href="#on_method_instance">on_method_instance</a>
<li>            <a href="#v_method_list">v_method_list</a>
<li>            <a href="#vertex_normal">vertex_normal</a>
<li>            <a href="#dihedral,-vertex">dihedral</a>
<li>            <a href="#mean_curvature,-vertex">mean_curvature</a>
<li>            <a href="#sqcurve,-vertex">square mean curvature</a>
<li>            <a href="#mid_edge">mid_edge</a>
<li>            <a href="#mid_facet">mid_facet</a>
<li>            <a href="#axial_point">axial_point</a>
<li>            <a href="#triple_point">triple_point</a>
<li>            <a href="#tetra_point">tetra_point</a>
<li>            <a href="#v_force">v_force</a>
<li>            <a href="#v_velocity">v_velocity</a>
<li>            <a href="#raw_velocity">raw_velocity</a>
<li>            <a href="#v_oldx">v_oldx</a>
</ul>
</td></tr></table>
<a href="#vertex-attributes">Full descriptions</a> of vertex attributes.
<hr>

<a   id="edges"></a><h2>Edges</h2>

            An edge is a one-dimensional <a href="#geometric-elements">
geometric element</a>.
  In the <a href="model.htm#linear-model">linear model</a>,
 an edge is an oriented line segment between a tail
            <a href="#vertices">vertex</a> and a head vertex. 
  In the <a href="model.htm#quadratic-model">quadratic model</a>, an edge is
defined by quadratic intepolation of two endpoints and a midpoint.
  In the <a href="model.htm#Lagrange-model">lagrange model</a>, an edge is
defined by the appropriate order interpolation with the edge vertices.
In the <a href="model.htm#string-model">string model</a>, edges carry
a default surface tension energy proportional to their length.
Edges may also carry energy by being on 
<a href="constrnt.htm#level-set-constraints">level set constraints</a>
in the 
<a href="model.htm#soapfilm-model">soapfilm model</a>,
 or by having <a href="quants.htm">
named quantity</a>  energies applied to them.
The edges of the original surface are defined in the 
<a href="datafile.htm#edges-section">edges section</a> of the datafile.
<p>
            Edge attributes (grouped somewhat relatedly>:
<table><tr><td style="vertical-align:text-top">
<ul>
<li>            <a href="#id">id</a>
<li>            <a href="#oid">oid</a>
<li>            <a href="#original">original</a>
<li>            <a href="#length">length</a>
<li>            <a href="#density,-edge">density or tension</a>
<li>            <a href="#fixed,-edge">fixed</a>
<li>            <a href="#edge-vertices">vertices</a>
<li>            <a href="#midv">midv</a>
<li>            <a href="#edge-facets">facets</a>
<li>            <a href="#valence,-edge">valence</a>
<li>            <a href="#bare,-edge">bare</a>
<li>            <a href="#constraints,-edge">constraints</a>
<li>            <a href="#on_constraint">on_constraint</a>
<li>            <a href="#e_constraint_list">e_constraint_list</a>
<li>            <a href="#boundary,-edge">boundary</a>
<li>            <a href="#on_boundary">on_boundary</a>
<li>            <a href="#e_boundary">e_boundary</a>
</ul></td><td style="vertical-align:text-top"><ul>
<li>            <a href="#color,-edge">color</a>
<li>            <a href="#edge-vector">edge_vector, x,y,z</a>
<li>            <a href="#no_refine,-edge">no_refine</a>
<li>            <a href="#no_transform,-edge">no_transform</a>
<li>            <a href="#wrap">wrap</a>
<li>            <a href="#wrap_list">wrap_list</a>
<li>            <a href="#show,-edge">show</a>
<li>            <a href="#orientation,-edge">orientation</a>
<li>            <a href="#frontbody,-edge">frontbody</a>
<li>            <a href="#backbody,-edge">backbody</a>
<li>            <a href="#dihedral,-edge">dihedral</a>
<li>            <a href="#noncontent,-edge">noncontent</a>
<li>            <a href="#quantity-attribute">quantities</a>
<li>            <a href="#on_quantity">on_quantity</a>
<li>            <a href="#quantity-attribute">method instances</a>
<li>            <a href="#on_method_instance">on_method_instance</a>
<li>            <a href="#e_method_list">e_method_list</a>
<li>            <a href="#extra-attributes">extra attributes</a>
</ul>
</td></tr></table>
<hr>
<a   id="facets"></a><h2>Facets</h2>
  In the <a href="model.htm#soapfilm-model">soapfilm model</a>,
  a facet is an oriented triangle defined by a cycle of three
  <a href="#edges">edges</a>.  
  In the <a href="model.htm#linear-model">linear model</a>,
  a facet is a flat triangle.
  In the <a href="model.htm#quadratic-model">quadratic
  model</a>, the facet is a curved surface defined by quadratic
  interpolation among the three facet corner vertices and the
  three edge midpoints.  In the <a href="model.htm#Lagrange-model">
  Lagrange model</a>, 
  <a href="syntax.htm#lagrange_order" class="keyword">lagrange_order</a>
   interpolation is done among
  (lagrange_order+1)(lagrange_order+2)/2 vertices.
  Although individual facets are oriented, there are no
  restrictions on the orientations of adjacent facets.
  By default, a facet carries a surface tension energy equal
  to its area.
  <p>
  In the <a href="model.htm#string-model">string model</a>,
  a facet is a chain of an arbitrary number of edges.  The chain
  need not be closed.  Usually a facet is defined in the string
  model in order to define a body, so the space dimension is 2
  and the facet is planar, one facet corresponding to a body.
  Facets carry no energy by themselves.
  <p>
  In the <a href="model.htm#simplex-model">simplex model</a>,
  a facet is a simplex of dimension
   <a href="datafile.htm#surface-dimension-decl" class="keyword">surface_dimension</a>
   defined by surface_dimension+1 vertices.  The <code>surface_dimension</code>
   may be any dimension less than or equal to the 
   <a href="datafile.htm#space-dimension-decl" class="keyword">space_dimension</a>.
   The simplex is oriented according to the order of the vertices.
   By default, a simplex carries a surface tension energy
   proportional to its volume.
<p>
  Facets may  carry additional energy by having
  <a href="quants.htm">named quantity</a> energies
  applied to them.
  <p>
  The facets of the original surface are defined in the 
  <a href="datafile.htm#faces-section">faces section</a> of the
  datafile.
  <p>
            Facet attributes (grouped somewhat relatedly):
<table><tr><td style="vertical-align:text-top">
<ul>
<li>            <a href="#id">id</a>
<li>            <a href="#oid">oid</a>
<li>            <a href="#original">original</a>
<li>            <a href="#density,-facet">density or tension</a>
<li>            <a href="#area,-facet">area</a>
<li>            <a href="#fixed,-facet">fixed</a>
<li>            <a href="#facet-vertices">vertices</a>
<li>            <a href="#facet-edges">edges</a>
<li>            <a href="#facet-bodies">bodies</a>
<li>            <a href="#frontbody,-facet">frontbody</a>
<li>            <a href="#backbody,-facet">backbody</a>
<li>            <a href="#f_body_list">f_body_list</a>
<li>            <a href="#valence,-facet">valence</a>
<li>            <a href="#constraints,-facet">constraints</a>
<li>            <a href="#on_constraint">on_constraint</a>
<li>            <a href="#f_constraint_list">f_constraint_list</a>
<li>            <a href="#boundary,-facet">boundary</a>
<li>            <a href="#on_boundary">on_boundary</a>
</ul></td><td style="vertical-align:text-top"><ul>
<li>            <a href="#f_boundary">f_boundary</a>
<li>            <a href="#color,-facet">color</a>
<li>            <a href="#frontcolor">frontcolor</a>
<li>            <a href="#backcolor">backcolor</a>
<li>            <a href="#opacity">opacity</a>
<li>            <a href="#facet_normal">facet_normal, x,y,z</a>
<li>            <a href="#no_display">no_display</a>
<li>            <a href="#no_refine,-facet">no_refine</a>
<li>            <a href="#no_transform,-facet">no_transform</a>
<li>            <a href="#orientation,-facet">orientation</a>
<li>            <a href="#noncontent,-facet">noncontent</a>
<li>            <a href="#phase,-facet">phase</a>
<li>            <a href="#quantity-attribute">quantities</a>
<li>            <a href="#on_quantity">on_quantity</a>
<li>            <a href="#quantity-attribute">method instances</a>
<li>            <a href="#on_method_instance">on_method_instance</a>
<li>            <a href="#f_method_list">f_method_list</a>
<li>            <a href="#extra-attributes">extra attributes</a>
<li>            <a href="#f_next_vfacet">f_next_vfacet</a>
<li>            <a href="#f_next_bfacet">f_next_bfacet</a>

</ul>
</td></tr></table>
<hr>
<a   id="bodies"></a><h2>Bodies</h2>

            A body is a full-dimensional region of space. Bodies
are not triangulated.  Rather, they are determined by their boundary
<a href="#facets">facets</a> (or <a href="#edges">edges</a> in 2D). 
                 These facets are used for calculating body volume
                 and gravitational energy.  Only those facets needed
                 for correct calculation need be given. In the 
                 <a href="model.htm#string-model">string
		 model</a>, usually a body corresponds to one facet.
		 Bodies of the original surface are defined in the
		 <a href="datafile.htm#bodies-section">bodies section</a>
		 of the datafile.  
<p>
            Body attributes:
<table><tr><td style="vertical-align:text-top">
<ul>
<li>            <a href="#id">id</a>
<li>            <a href="#original">original</a>
<li>            <a href="#body-facets">facets</a>
<li>            <a href="#density,-body">density</a>
<li>            <a href="#body-volume">volume</a>
<li>            <a href="#target-volume">target</a>
<li>            <a href="#volfixed">volfixed</a>
</ul></td><td style="vertical-align:text-top"><ul>
<li>            <a href="#body-volconst">volconst</a>
<li>            <a href="#actual_volume">actual_volume</a>
<li>            <a href="#pressure,-body">pressure</a>
<li>            <a href="#phase,-body">phase</a>
<li>            <a href="#centerofmass">centerofmass</a>
<li>            <a href="#extra-attributes">extra attributes</a>

</ul>
</td></tr></table>
<hr>
<a   id="facetedges"></a><h2>Facetedges</h2>
A facetedge is a pairing of a facet and one of its edges, with
orientation such that the edge orientation is consistent with
the facet orientation.  Facetedges are used internally by Evolver,
and are seldom of interest to the user.  They carry no energy.
The '<a href="single.htm#C" class="keyword">C</a>' 
integrity-checking command will sometimes refer to
facetedges if the surface is inconsistent.  <code>Facetedge</code> can
be used as an element generator.
<p>
            Facetedge attributes:
<ul>
<li> <a href="#id">id</a>
<li> <a href="#facetedge-edge">edge</a>
<li> <a href="#facetedge-facet">facet</a>
<li> <a href="#nextedge">nextedge</a>
<li> <a href="#prevedge">prevedge</a>
<li> <a href="#nextfacet">nextfacet</a>
<li> <a href="#prevfacet">prevfacet</a>
<li> <a href="#extra-attributes">extra attributes</a>
</ul>
<hr>
<hr>
<a   id="element-attributes"></a>
<h1>Element attributes</h1>
Below is a list of possible element attributes.  The first few apply to 
all types of elements.  Then come those applying specifically to vertices,
edges, facets, and bodies.  See <a href="#geometric-elements">
Geometric elements</a> for lists of attributes for each type element.
<hr>


<!--========================== Common attributes ===========================-->

<h1>Attributes for all types of elements</h1>
<hr>


<!--============================== id ============================-->

<a   id="id"></a><h2>id</h2>
<a href="#geometric-elements">Geometric element</a> read-only integer attribute.
The id of an element is a positive integer uniquely associated with
that element.  The Evolver will assign id's to elements read from
the datafile in the order they are read, unless the 
<a href="general.htm#options">-i command line option</a>
 or <a href="datafile.htm#keep_originals" class="keyword">keep_originals</a> 
is in the top of the datafile, in which
case the datafile element number is the id.
  In either case, you can access the datafile id with the <a href="#original">
original</a> attribute.  Examples:
<pre>   list vertex where id &lt; 10
   set edge color red where id == 4 or id == 6 or id == 9
   foreach facet ff do 
   { printf "%d  %d %d %d\n",ff.id,ff.edge[1].id,ff.edge[2].id,ff.edge[3].id } 
</pre>
<hr>

<!--============================== oid ============================-->

<a   id="oid"></a><h2>oid</h2>
<a href="#geometric-elements">Geometric element</a> read-only integer attribute.
The oid of an element is the "oriented id" of an element as used in
an expression.  It is the <a href="#id">id</a> number signed according
to whether the use of the element is with the same or opposite orientation
as the way it is stored.  Example: to get an edge list for a facet
as in the datafile, use oid instead of id:
<pre>
   foreach facet ff do { printf "%d  %d %d %d\n",ff.id,ff.edge[1].oid,
      ff.edge[2].oid,ff.edge[3].oid } 
</pre>

<hr>

<!--============================== fixed ============================-->

<a   id="fixed"></a><h2>fixed</h2>
<a href="#geometric-elements">Geometric element</a> read-write boolean attribute.
For vertices, <code>fixed</code> means they don't move during various evolution
and triangulation grooming commands.  For edges and facets, <code>fixed</code> 
means vertices generated from them by refinement
 are <code>fixed</code>, although declaring 
a facet or edge fixed does not automatically make its vertices <code>fixed</code>.
<p>
For a body, <code>fixed</code> means its volume is constrained to be its
<a href="#target">target</a> value. Likewise, <code>fixed</code> as an attribute
of a <a href="quants.htm#named-quantities">named quantity</a> means the quantity
value is constrained.
<p>
Fixedness can be changed with the <a href="commands.htm#fix" class="keyword">
fix</a> and <a href="commands.htm#unfix" class="keyword">unfix</a> commands.
<p>
Examples:
<pre>
   fix vertex where z = 0
   unfix vertex where on_constraint 1
   list edge where fixed
   print body[1] fixed
</pre>



<hr>

<!--============================== on_constraint ============================-->

<a   id="on_constraint"></a><h2>on_constraint</h2>
<a href="#vertices">Vertex</a>,
<a href="#edges">edge</a>, or
<a href="#facets">facet</a> 
read-only attribute.
<a href="syntax.htm#boolean-ops">Boolean</a> attribute for whether an element is on a given
<a href="constrnt.htm#level-set-constraints">constraint</a>.
The full syntax of the attribute is "<code>on_constraint</code> <em>n</em>"
where <em>n</em> is the number or name of the constraint. Examples:
<pre>
   list edge where on_constraint 3
   print vertex[3].on_constraint floorcon
</pre>

<hr>

<!--============================== on_boundary ============================-->

<a   id="on_boundary"></a>
<h2>on_boundary</h2>
<a href="#vertices">Vertex</a>,
<a href="#edges">edge</a>, or
<a href="#facets">facet</a> 
read-only boolean attribute.

The status of whether an element is on a 
<a href="constrnt.htm#parametric-boundaries">parametric boundary</a> 
The full syntax of the attribute is "<code>on_boundary </code><em>n</em>" where
<em>n</em> is the number or name of the boundary. 
Examples:
<pre>
  list vertex where on_boundary 1
  list edge where on_boundary topbdry
</pre>


<hr>

<!--============================== on_quantity ============================-->

<a   id="on_quantity"></a><h2>on_quantity</h2>
<a href="#vertices">Vertex</a>,
<a href="#edges">edge</a>, or
<a href="#facets">facet</a> 
read-only attribute.
<a href="syntax.htm#boolean-ops">Boolean</a> attribute for whether an element
contributes to a given
<a href="quants.htm#named-quantities">named quantity</a>.
Actually, it tests whether the element is on any of the method instances
comprising a quantity.
The full syntax of the attribute is 
"<code>on_quantity</code> <em>quantityname</em>".
 Examples:
<pre>
   list facet where on_quantity  center_of_mass_x
   print vertex[3].on_quantity blue_area
</pre>


<hr>

<!--============================== on_method_instance ============================-->

<a   id="on_method_instance"></a><h2>on_method_instance</h2>
<a href="#vertices">Vertex</a>,
<a href="#edges">edge</a>, or
<a href="#facets">facet</a> 
read-only attribute.
<a href="syntax.htm#boolean-ops">Boolean</a> attribute for whether an element
contributes to a given
<a href="quants.htm#named-methods">named method instance</a>.
The full syntax of the attribute is 
"<code>on_method_instance</code> <em>instancename</em>".
 Examples:
<pre>
   list facet where on_method_instance  center_of_mass_x_edges
   print vertex[3].on_method_instance blue_area_1
</pre>



<hr>

<!--============================== original ============================-->

<a   id="original"></a><h2>original</h2>
<a href="#geometric-elements">Geometric element</a> read-write integer attribute.
For elements read from the datafile, this is the number given to the
element in the datafile, which may be overridden by an explicit original
attribute value in the datafile line defining the element.  The value is
inherited by all elements of the same type that result from subdivision.
For elements otherwise
generated at run time, the original attribute value is -1.
Example: to show which facets descended from face 1 in the datafile:
<pre>
   set facet color red where original == 1
</Pre>

<hr>
<!--============================== named quantity ============================-->

<a   id="quantity-attribute"></a><h2>Named quantity values as attributes</h2>

<a href="quants.htm">Named quantities</a> and
<a href="quants.htm">method instances</a> can be applied to geomtric
elements either in the <a href="datafile.htm">datafile</a> (by
adding the quantity or method name to the line defining an element) or
with the <a href="commands.htm#set">set</a> command.
 Nonglobal quantities
or methods can be <a href="commands.htm#unset">unset</a>
 for individual elements. The values for individual
elements can be accessed using attribute syntax. 
Examples: Suppose there is a named quantity "xmoment" that can be 
evaluated for facets.  Then one could give commands
<pre>
   foreach facet do printf "%d %f\n",id,xmoment
   list facet where xmoment &gt; 4
</pre>

<hr>
<!--============================== Extra attributes ============================-->

<a   id="extra-attributes"></a><h2>Extra attributes</h2>

<a href="#geometric-elements">Geometric element</a> read-write attributes.
If <a href="elements.htm#extra-attributes">extra attributes</a>
 have been defined in the
<a href="datafile.htm#extra-decl">datafile</a> or with a 
<a href="commands.htm#define">define</a> command,
they can be accessed with attribute syntax. Extra attribute
values in the datafile can be initialized for an element
by adding the attribute name and value to the line defining the
element.  Extra attributes may also be arrays of numeric values 
(global arrays can be strings, but attributes cannot yet), initialized
with standard nested bracket syntax. Example:
<pre>
  define vertex attribute newx real
  define vertex attribute vmat real[3][2]
  vertices
  1   2 0 0 newx 3 vmat {{1,2},{3,4},{5,6}}
</pre>
The command language can use the name with the same syntax as built-in
attributes, and can define extra attributes at run time:
<pre>
  set vertex newx x
  define edge attribute vibel real[2]
  set edge[2] vibel[1] 3; set edge[2] vibel[2] 4
  print vertex[3].newx
</pre>
Attribute array sizes may be changed at run time by executing another
definition of the attribute, but the number of dimensions must be the same.
Array entry values are preserved as far as possible when sizes are
changed.
<p>
The value of an extra attribute can also be calculated by user-supplied
code. The attribute definition is followed by the keyword "function" and
then the code in brackets. In the code, the keyword "self" is used
to refer to the element the attribute is being calculated for. 
Example: To implement the lowest z value of a facet as an attribute:
<pre> define facet attribute minz real function
	 {self.minz := min(self.vertex,z);}
</pre>

<!--============================== Vertex attributes ============================-->

<hr>
<hr>
<a   id="vertex-attributes"></a>
<h1>Vertex attributes</h1>

<!--============================== Vertex id ============================-->

<hr>
<H2>Vertex id</h2> See <a href="#id" class="keyword">id</a> for general elements.

<!--============================== Vertex original ============================-->

<hr>
<H2>Vertex original</h2> See <a href="#original" class="keyword">original</a> for general elements.


<hr>
<!--============================== Vertex coordinates ============================-->

<a   id="coordinates"></a>
<a   id="x1"></a>
<a   id="x2"></a>
<a   id="x3"></a>
<a   id="x-coordinate"></a><a   id="y-coordinate"></a>
<a   id="z-coordinate"></a><a   id="w-coordinate"></a>
<a   id="__x"></a>
<h2>Vertex coordinates: x,y,z, x[], __x[]</h2>
<a href="#vertices">Vertex</a> read-write real attribute. The coordinates of
a vertex are its location in space.  By default, these are Euclidean
coordinates, but they may represent any coordinate system if the
user defines appropriate length, area, volume, etc. integrals.
But graphics always treat the coordinates as Euclidean.  The
individual coordinates may be referred to as <code>x</code>,<code>y</code>,<code>z</code>,
<code>w</code> or <code>x1</code>,<code>x2</code>,<code>x3</code>,..,
or <code>x[1]</code>, <code>x[2]</code>, <code>x[3]</code>, ... or
 <code>__x[1]</code>, <code>__x[2]</code>, <code>__x[3]</code>, .... 
In the <a href="datafile.htm#vertices-section">vertices section</a>
of the datafile,
vertices of the original surface have their coordinates given 
 unless
they are on a <a href="constrnt.htm#parametric-boundaries"> parametric
boundary</a>.  Vertices on parametric boundaries have their coordinates
calculated from their parameter values.  Coordinates may be read or
modified with the command language.  The form <code>__x</code> is useful to refer
to the coordinates as a vector, for example in dot products.
Examples:
<pre>
  foreach vertex do printf "%d  %f %f %f\n",id,x,y,z
  set vertex z z+.1*x
  print vertex[3].x[2]
  dotprod := vertex[1].__x * vertex[2].__x
</pre>
<hr>
<!--============================== Vertex edges ============================-->

<a   id="vertex-edges"></a><h2>Vertex edges</h2>
<a href="#vertices">Vertex</a> read-only generator attribute. 
<a href="commands.htm#generators">Generates</a>
edges
attached to a vertex, oriented so vertex is the edge tail. The edges
are in no particular order.
Examples:
<pre>
  list vertex[3].edges
  foreach vertex vv do { foreach vv.edge do print id }
</pre> 
Always use "<code>.edges</code>" to generate vertex edges; using "edges" with
an implicit element, as in "<code>foreach vertex do list edges</code>"
will list all edges in the surface over and over again.
<hr>
<!--============================== Vertex facets ============================-->

<a   id="vertex-facets"></a><h2>Vertex facets</h2>
<a href="#vertices">Vertex</a> read-only generator attribute. 
<a href="commands.htm#generators">Generates</a>
facets
attached to a vertex, with positive facet orientation. The facets are
in no particular order.
Examples:
<pre>
  list vertex[3].facets
  foreach vertex vv do { foreach vv.facet do print id }
</pre>
Always use "<code>.facets</code>" to generate vertex facets; using "facets" with
an implicit element, as in "<code>foreach vertex do list facets</code>"
will list all facets in the surface over and over again.
<hr>
<!--============================== Vertex valence ============================-->

<a   id="valence,-vertex"></a><h2>Vertex valence</h2>
<a href="#vertices">Vertex</a> read-only integer attribute. 
The valence of a vertex is defined to be the number of edges
it is a member of.  Example:
<pre>
  list vertices where valence == 6
  histogram(vertex,valence)
</pre>

<hr>
<!--============================== Vertex bare ============================-->

<a   id="bare,-vertex"></a><h2>Vertex bare</h2>
<a href="#vertices">Vertex</a> read-write boolean attribute. 
Declaring a vertex <code>bare</code> says that a vertex is not supposed to have 
any adjacent edges.
  Useful in avoiding warning messages.  A vertex may be declared bare
  in the <a href="datafile.htm#vertices-section">vertices section</a>
  of the datafile by adding the keyword <code>bare</code> to the line
  defining the vertex. <code>Bare</code> is not simply a synonym for zero
<a href="#valence,-vertex" class="keyword">valence</a>; it is a separate attribute
you set to say you intend for it to have zero valence.
Examples:
<pre>
   set vertex bare where valence==0
   list vertex where bare
</pre>
<hr>  
<!--============================== Vertex fixed ============================-->

<a   id="fixed,-vertex"></a><h2>Vertex fixed</h2>
<a href="#vertices">Vertex</a> read-write boolean attribute.
A fixed vertex will not move during iteration (except to satisfy
<a href="constrnt.htm#level-set-constraints">level set constraints</a>)
 or other operations, except if coordinates
are explicitly changed by a "<a href="commands.htm#set">set vertices ...</a>"
command.  
A vertex may be declared fixed in the datafile
by putting <code>fixed</code> on the line defining the vertex, after the
coordinates.  From the command prompt, one can fix or unfix vertices
with the <a href="commands.htm#fix" class="keyword">fix</a> and
<a href="commands.htm#unfix" class="keyword">unfix</a> commands.
Examples:
  formula
<pre>
  list vertex where fixed
  fix vertex where on_constraint 1
  unfix vertices where on_boundary 1
</pre>

<hr>
<!--============================== Vertex constraints ============================-->

<a   id="constraints,-vertex"></a><h2>Vertex constraints</h2>
<a href="#vertices">Vertex</a> read-write attribute. 
          A 
<a href="constrnt.htm#level-set-constraints">level-set constraint</a>
 is a restriction of vertices to
lie on the zero level-set of a function.  A constraint declared
<code>nonnegative</code> in the 
<a href="datafile.htm#constraint-decl">datafile</a>
 forces a vertex to have a nonnegative
value of the function.  A <code>nonpositive</code> constraint forces a vertex
to have a nonpositive value of the function.
A constraint may be declared <code>global</code>,
          in which case it applies to all vertices.  A vertex may
    be put on a constraint in the <a href="datafile.htm#vertices-section">
vertices section</a>
    of the datafile by listing the constraint numbers after the keyword
    <code>constraint</code>.
	  See  mound.fe for an example.
In commands, the status of a vertex can be read with the 
<a href="#on_constraint" class="keyword">on_constraint</a> and 
<a href="#hit_constraint" class="keyword">hit_constraint</a>
attributes.  The status can be changed with the 
<a href="commands.htm#set" class="keyword">set</a> or 
<a href="commands.htm#unset" class="keyword">unset</a>
 commands.  
Examples:
<br>  Datafile:
<pre>
  constraint 1
  formula: z = 0
  constraint wallcon nonpositive
  formula: x = 4
</pre>
Runtime:
<pre>
  list vertex where on_constraint wallcon
  set vertex constraint 1 where id == 4 or id == 6
  unset vertex constraint 1
</pre>

<!--============================== Vertex on_constraint ============================-->
<hr>
See the <a href="#on_constraint" class="keyword">on_constraint</a>
attribute for general elements.

<!--============================== Vertex hit_constraint ============================-->

<hr>
<a   id="hit_constraint"></a><h2>Vertex hit_constraint</h2>
<a href="#vertices">Vertex</a> read-only attribute.
<a href="syntax.htm#boolean-ops">Boolean</a> attribute for whether a vertex exactly satisfies a given
<a href="constrnt.htm#level-set-constraints">constraint</a>.  Particularly
meant for vertices on 
<a href="constrnt.htm#one-sided-constraints">one-sided constraints</a>.
The full syntax of the attribute is "<code>hit_constraint</code> <em>n</em>"
where <em>n</em> is the number or name of the constraint. Examples:
<pre>
   list vertex where hit_constraint 3
   print vertex[3].hit_constraint 1
</pre>

<!--============================== Vertex value_of_constraint ============================-->

<hr>
<a   id="value_of_constraint"></a><h2>Vertex value_of_constraint</h2>
<a href="#vertices">Vertex</a> read-only attribute giving the value
of a  <a href="constrnt.htm#level-set-constraints"> level-set constraint</a>
for the current position of a vertex. 
 Particularly meant for vertices on 
<a href="constrnt.htm#one-sided-constraints">one-sided constraints</a>.
The full syntax of the attribute is "<code>value_of_constraint</code> <em>n</em>"
where <em>n</em> is the number or name of the constraint. Examples:
<pre>
   print vertex[4] value_of_constraint 1
   list vertex where on_constraint topcon and value_of_constraint topcon > 1e-5
</pre>

<!--============================== Vertex v_constraint_list ============================-->

<hr>
<a   id="v_constraint_list"></a><h2>Vertex v_constraint_list</h2>
This read-only integer array attribute gives access to the list of constraints a
vertex is on. <code>v_constraint_list[1]</code> is the number of constraints in
the list, followed by the numbers of the constraints.  Note that for
named constraints, the internally assigned numbers are used.  Because this is
the actual internal datastructure, the entries may have some high bits used
as flags, so to get the plain constraint numbers you should mask out the 
high bits with "<code class="shortcode">imod 0x100000</code>".
Example:
<pre>
   foreach vertex vv do
   { for ( spot := 2 ; spot &lt;= vv.v_constraint_list[1]+1  ; spot++ )
     { connum := vv.v_constraint_list[spot] imod 0x100000;
       if vv hit_constraint connum then
         printf "Vertex %d hits constraint %d.\n",vv.id,connum;
     }
   };
</pre>

<!--============================== constraint normal ============================-->
<hr>
<a   id="constraint-normal"></a>
<h2>Constraint normal</h2>

The unit normal vector of a level-set constraint at a vertex may be
found with the vertex vector attribute
<pre>        constraint[number].normal
or      constraint[name].normal </pre>
"number" may be an expression; "name" is the unquoted name of the
constraint, if it has one.  Example: <pre>
  print vertex[1].constraint[floorcon].normal
</pre>
would print the unit normal of constraint floorcon at vertex 1.  And
you can put on a subscript to get individual components.  For example,
the y component: <pre>
  print vertex[1].constraint[floorcon].normal[2]
</pre>
There is no necessity for the constraint to be applied to the vertex;
the vertex is just used as a source of coordinates for evaluating the
gradient of the constraint formula.

<!--============================== Vertex boundary ============================-->
<hr>
<a   id="boundary,-vertex"></a>
<h2>Vertex boundary</h2>
<a href="#vertices">Vertex</a> read-write attribute.  A vertex may be on 
a <a href="constrnt.htm#boundary">parameterized boundary</a>, in which case
its position is specified by parameter values; i.e. the space coordinates
are functions of the parameters. The keyword <code>boundary</code> is used as
an attribute only in the datafile declaration of elements being on boundaries.
At runtime, one uses the attributes 
<a href="#parameter-values" class="keyword">p1, p2, ..</a>
for the parameter values, <a href="#on_boundary" class="keyword">on_boundary</a>
to see if a vertex is on a particular boundary, and <a href="#v_boundary"
class="keyword">v_boundary</a> to get the number of the boundary.  If you
want to set a vertex on a boundary at runtime (tricky, since you have to set 
the parameters yourself), the use the 
"<a href="commands.htm#set" class="shortcode">set vertex boundary ...</a>" command.
<hr>


<!--============================== Vertex parameters ============================-->

<a   id="p1"></a><a id="p2"></a>
<a   id="parameter-values"></a><h2>Vertex parameters p1, p2, p[]</h2>
<a href="#vertices">Vertex</a> read-write real attribute.
 Vertices on parametric boundaries are located according to the
parameter values.  Parameters are referred to as <code>p1</code>,<code>p2</code>,...  Usually
only <code>p1</code> is used, since one-parameter curves used as boundary wires
are most common.   There is also an array form of the name, <code>p</code>, which
is useful in array computations such as dot product in the case of
multiple parameters.  Such vertices in the original surface have
their parameter values given in the 
<a href="datafile.htm#vertices-section">vertices section</a> of the datafile
instead of their coordinates. Vertex parameters may be read or modified
with the command language. Example:
<pre>
  foreach vertex do printf "%d %f\n",id,p1
  set vertex[1] p1 1.2
  dotprod := vertex[1].p * vertex[2].p
</pre>

<hr>
<!--============================= vertex on_boundary =========================-->

<h2>Vertex on_boundary</h2>
See <a href="#on_boundary" class="keyword">on_boundary</a> for general elements.


<!--============================== Vertex v_boundary ============================-->

<hr> <a   id="v_boundary"></a><h2>Vertex v_boundary</h2>
<a href="#vertices">Vertex</a> read-only integer attribute.  Internal attribute
containing the number of any parameterized boundary the vertex is on.
Recall that named boundaries have internal id numbers, which are used here.

<!--============================= vertex extra attributes =========================-->
<hr>
<h2>Vertex extra attributes</h2>
See <a href="#extra-attributes">extra attributes</a> for general elements.

<!--============================= vertex named quantity =========================-->
<hr>
<h2>Vertex named quantity</h2>
See <a href="#quantity-attribute">named quantities</a> for general elements.

<!--============================= vertex on_quantity =========================-->
<hr>
<h2>Vertex on_quantity</h2>
See <a href="#on_quantity">on_quantity</a> for general elements.


<!--======================= vertex named method instances =========================-->
<hr>
<h2>Vertex named method instances</h2>
See <a href="#quantity-attribute">named quantities</a> for general elements.

<!--============================= vertex on_method_instance =========================-->
<hr>
<a   id="vertex-on_method_instance"></a><h2>Vertex on_method_instance</h2>
See <a href="#on_method_instance">on_method_instance</a> for general elements.

<!--============================== Vertex v_method_list ============================-->

<hr> <a   id="v_method_list"></a><h2>Vertex v_method_list</h2>
<a href="#vertices">Vertex</a> read-only integer array attribute.  Internal name
for the array holding the id numbers of the method instances
to which this vertex contributes. Vertices do not directly record which quantities
they are on, they only record which method instances.
<hr>

<!--============================== Vertex vertex_normal ============================-->

<a   id="vertexnormal"></a>
<a   id="vertex_normal"></a><h2>Vertex vertex_normal or vertexnormal </h2>
<a href="#vertices">Vertex</a> read-only real array attribute.  
One-dimensional, size is space dimension. This is an indexed
attribute consisting of the components of a normal to the surface at
a vertex, normalized to unit length.  This is the same normal as used
in <a href="toggle.htm#hessian_normal">hessian_normal</a> mode.  For
most vertices in the soapfilm model, the normal is the number average
of the unit normals of the surrounding facets.  Along triple edges
and such where hessian_normal has a multi-dimensional normal plane,
the vertex_normal is the first basis vector of the normal plane.
Example: To print the normal components of vertex 3:
<pre> print vertex[3].vertex_normal[1];
 print vertex[3].vertex_normal[2];
 print vertex[3].vertex_normal[3]; </pre>
The vertex_normal can also be printed as an array:
<pre> 
   print vertex[3].vertex_normal 
</pre>

<code>Vertexnormal</code> is an old synonym for <code>vertex_normal</code>.
<hr>
<!--============================== Vertex dihedral ============================-->


<a   id="dihedral,-vertex"></a>
<h2>Vertex dihedral</h2>

<a href="#geometric-elements">Vertex</a> read-only real attribute in the string 
model.  This is the angle in radians from straightness of two edges at a vertex.
If there are less than two edges, the value is 0.  If two or more
edges, the value is 2*asin(F/2), where F is the magnitude of the net
force on the vertex, assuming each edge has tension 1.  Upper limit
clamped to pi.
<hr>
<!--============================== Vertex mid_edge ============================-->

<a   id="mid_edge"></a><h2>Vertex mid_edge</h2>
<a href="#vertices">Vertex</a> read-only boolean attribute.  True (1) if the
vertex is on an edge but not an endpoint.  Relevant in the 
<a href="model.htm#quadratic-model">quadratic model</a> or
<a href="model.htm#Lagrange-model">Lagrange model</a>.
Example:
<pre>
   list edge[23].vertex vv where vv.mid_edge
</pre>

<hr>
<!--============================== Vertex mid_facet ============================-->

<a   id="mid_facet"></a><h2>Vertex mid_facet</h2>
<a href="#vertices">Vertex</a> read-only boolean attribute.  True (1) if the
vertex is an interior control point of a facet in the
<a href="model.htm#Lagrange-model"><b>Lagrange model</b></a>.
Example:
<pre>
   list facet[23].vertex vv where vv.mid_facet
</pre>


<hr>
<!--============================== Vertex mean_curvature ============================-->

<a   id="mean_curvature,-vertex"></a><h2>Vertex mean_curvature</h2>
<a href="#vertices">Vertex</a> read-only ral attribute, available in the
string and soapfilm model. The mean curvature is calculated as the 
magnitude of the gradient of area (or length in the string model)
divided by the area (or length) associated with the vertex, which is
one-third the area of the facets adjacent to the vertex (or one-half of
the length of adjacent edges).  It is divided by 2 in the soapfilm model
to account for the "mean" part of the definition.  The sign of the
mean curvature is relative to the orientation of the first adjacent
facet (or edge) Evolver finds. This calculation can be done even if the 
vertex is on a triple junction or other non-planar topology, even if it 
doesn't interpret well as mean curvature there. 

<hr>
<!--============================== Vertex sqcurve ============================-->

<a   id="sqcurve,-vertex"></a>
<h2>Vertex sqcurve</h2>

<a href="#geometric-elements">Geometric element</a> read-only real attribute.
<code>Sqcurv</code>
 is the squared mean curvature at a vertex.  This is only a discrete 
approximation, of course, The method used to calculate it is the 
same as the <a href="quants.htm#sq_mean_curvature" class="keyword">sq_mean_curvature</a>
named method, except if the <a href="toggle.htm#normal_curvature" class="keyword">
normal_curvature</a> toggle is on, in which case the calculation is as in
the <a href="quants.htm#normal_sq_mean_curvature" class="keyword">
normal_sq_mean_curvature</a> named method.  Does not require any other square
mean curvature features to be active.
<hr>
<!--============================== Vertex axial_point ============================-->

<a   id="axial_point"></a><h2>Vertex axial_point</h2>
<a href="#vertices">Vertex</a> read-write boolean attribute.
Certain <a href="model.htm#symmetry-groups">symmetry groups</a>
(e.g. <a href="model.htm#cubocta">cubocta</a> or
<a href="model.htm#rotate-symmetry-group">rotate</a>)
have axes of rotation that are invariant under some non-identity
group element.  A vertex on such an axis must be labeled in the
datafile with the attribute <code>axial_point</code>, since these
vertices pose special problems for the wrap algorithms.
If you are only using a subgroup of the full group, then you
only need to label vertices on the axes of the subgroup.
The net wrap around a facet containing an axial point need not
be the identity. Edges out of an
axial point must have the axial point at their tail, and must have zero 
wrap.  Facets including an axial point must have the axial point at
the tail of the first edge in the facet.  It is your responsibility 
to use constraints to guarantee the vertex remains on the axis.


<hr>
<!--============================== Vertex triple_point ============================-->

<a   id="triple_point"></a><h2>Vertex triple_point</h2>
<a href="#vertices">Vertex</a> read-write boolean attribute. For telling Evolver
three films meet at this vertex.  Used when effective_area is on to
adjust motion of vertex by making the effective area around the vertex
1/sqrt(3) of actual.
<hr>
<!--============================== Vertex triple_point ============================-->

<a   id="tetra_point"></a><h2>Vertex tetra_point</h2>
<a href="#vertices">Vertex</a> read-write booloean attribute. For telling Evolver
six films meet at this vertex.  Used when effective_area is on to
adjust motion of vertex by making the effective area around the vertex
1/sqrt(6) of actual.


<hr>
<!--============================== Vertex v_force ============================-->

<a   id="v_force"></a><h2>Vertex v_force</h2>
<a href="#vertices">Vertex</a> read-only real array attribute.  This is an indexed
attribute giving the components of the force (negative energy
gradient as projected to constraints). One-dimensional, size is space dimension.
 Meant for debugging use.
This is not directly used for the motion; see <a href="#v_velocity">
v_velocity</a>. 
<hr>

<!--============================== Vertex v_velocity ============================-->

<a   id="v_velocity"></a><h2>Vertex v_velocity</h2>
<a href="#vertices">Vertex</a> read-only real array attribute.  This is an indexed
attribute giving the components of the vector used for vertex motion
in the '<a href="single.htm#g" class="keyword">g</a>' command.  
The motion of a vertex is the <a href="iterate.htm#scale-factor">scale factor</a>
times this vector.  The velocity vector is calculated from the force vector
by applying area normalization, mobilty, etc.  Also, if a vertex is
on a boundary, the velocity is projected back to parameters. 
One-dimensional, size is space dimension.


<hr>
<!--============================== Vertex raw_velocity ============================-->

<a   id="raw_velocity"></a><h2>Vertex raw_velocity</h2>
<a href="#vertices">Vertex</a> read-only real array attribute. Internal vertex attribute 
used when one-sided level-set constraints are present, so the Lagrange multipliers 
for said constraints can be calculated.  This is the velocity before any projection 
to volume or level-set constraints.  One-dimensional, size is space dimension.
Not of interest to the ordinary user.
 
<!--============================== Vertex v_oldx ============================-->

<hr> <a   id="v_oldx"></a><h2>Vertex v_oldx</h2>
<a href="#vertices">Vertex</a> read-only real array attribute. 
Internal vertex array attribute used to store old coordinates when
doing an optimizing move. One-dimensional, size is space dimension.


<!--============================== Vertex no_hessian_normal ============================-->

<a   id="no_hessian_normal"></a><h2>Vertex no_hessian_normal</h2>
<a href="#vertices">Vertex</a> read-write attribute.
If you wish to run in hessian_normal mode but exempt particular vertices
from the restriction, you can "set" the vertices' no_hessian_normal
attribute, for example
<pre>   set vertex no_hessian_normal where z > 1.2 </pre>





<!--============================== Edge attributes ============================-->


<hr>
<hr>
<h1>Edge attributes</h1>

<hr>
<!--============================== Edge id ============================-->

<H2>Edge id</h2> See <a href="#id" class="keyword">id</a> for general elements.

<hr>
<!--============================== Edge oid ============================-->

<H2>Edge oid</h2> See <a href="#oid" class="keyword">oid</a> for general elements.

<!--============================== Edge original ============================-->

<hr>
<H2>Edge original</h2> See <a href="#original" class="keyword">original</a> 
for general elements.


<hr>
<!--============================== Edge length ============================-->

<a   id="length"></a>
<h2>Edge length</h2>
<a href="#edges">Edge</a> read-only real attribute. Length of the edge.
Examples:
<pre> histogram(edge where on_constraint 1, length)
 print edge[3].length
</pre>
<hr>
<!--============================== Edge tension ============================-->

<a   id="density,-edge"></a><a   id="tension,-edge"></a>
<h2>Edge density or tension</h2>
<a href="#edges">Edge</a> read-write real attribute.  
"Density" and "tension" are synonyms. 
Energy per unit
length of edge. Default 1 in string model, 0 in soapfilm model.
The tension may be modified in the datafile 
<a href="datafile.htm#edges-section">edges section</a> by
adding "<code>tension</code> <em>value</em>" to the line defining the edge.
The tension may be modified with the <a href="commands.htm#set">set</a>
command.
Examples:
<pre>  set edge tension .5 where id &lt; 10
  loghistogram(edge,density)
</pre>
<hr>
<!--============================== Edge fixed ============================-->

<a   id="fixed,-edge"></a><h2>Fixed edge</h2>
<a href="#edges">Edge</a> read-write attribute.
For an edge to be "fixed" means that any vertex or edge created
by refining the edge will inherit the "fixed" attribute.
Declaring an edge fixed in the datafile will not fix vertices
on the edge, and fixing an edge from the command prompt
will not fix any vertices.
An edge may be declared fixed in the datafile 
<a href="datafile.htm#edges-section">edges section</a>
by adding <code>fixed</code> to the line defining the edge.
 From the command prompt, one can fix or unfix edges
with the <a href="commands.htm#fix" class="keyword">fix</a> and
<a href="commands.htm#unfix" class="keyword">unfix</a> commands.
Examples:
<pre>  fix edge where on_constraint 1
  list edges where fixed
  set edge color red where fixed
  unfix edge[3]
</pre>
<hr>
<!--============================== Edge vertices ============================-->

<a   id="edge-vertices"></a><h2>Edge vertices</h2>
<a href="#edges">Edge</a> read-only attribute. Acts as a
<a href="commands.htm#generators">generator</a>
 for the vertices on an edge. In 
the <a href="model.htm#linear-model">linear model</a>, this
means the tail and head vertices. In the
<a href="model.htm#quadratic-model">quadratic</a> model,
it means the tail, head, and middle vertices. 
For the <a href="model.htm#Lagrange-model">Lagrange</a> model,
it means all the vertices from tail to head in order. 
For the <a href="model.htm#simplex-model">simplex</a> model,
it means the vertices in the stored order.
Example:
<pre>
   list edge[2].vertices
   list edge ee where ee.vertex[1].on_constraint 1 
</pre>
<hr>
<!--============================== Edge midv ============================-->

<a   id="midv"></a><h2>Edge midv</h2>
<a href="#edges">Edge</a> read-only attribute.
In the <a href="model.htm#quadratic-model">quadratic model</a>, gives
the id of the midpoint vertex of an edge.  Example:
<pre>  print edge[23].midv </pre>
<hr>

<!--============================== Edge facets ============================-->
<a   id="edge-facets"></a><h2>Edge facets</h2>
<a href="#edges">Edge</a> read-only attribute. 
<a href="commands.htm#generators">Generates</a>
facets
attached to an edge, in order around the edge when meaningful,
 with facet orientation agreeing with edge orientation. 
Examples:
<pre>
   list edge[2].facets
   foreach edge ee do print max(ee.facets,area)
</pre>
<hr>
<!--============================== Edge valence ============================-->

<a   id="valence,-edge"></a><h2>Edge valence</h2>
<a href="#edges">Edge</a> read-only integer attribute.
The valence of an edge is the number of facets adjacent to it.
Examples:
<pre>  list edges where valence == 1
  refine edge where valence != 2
</pre>
<hr>

<!--============================== Edge bare ============================-->

<a   id="bare,-edge"></a><h2>Bare edge</h2>
<a href="#edges">Edge</a> read-write boolean attribute. 
Declaring an edge "bare" indicates that an edge does not have an
adjacent facet (soapfilm model).  Best declared in the
<a href="datafile.htm#edges-section">datafile</a>, by adding the
keyword <code>bare</code> to the line defining an edge.
  Useful in avoiding warning
messages.   Bare edges are useful to show wires, frameworks, outlines, axes,
etc. in graphics. Example:
<pre>
  list edge where bare
</pre>
<hr>
<!--============================== Edge constraints ============================-->

<a   id="constraints,-edge"></a><h2>Edge constraints</h2>
<a href="#edges">Edge</a> read-write attribute.
An edge may be put on a <a href="constrnt.htm#level-set-constraints">
level set constraint</a>.
For such an edge,
any vertices and edges generated
by refining the edge will inherit the constraint.  An edge may
be put on constraints in the 
<a href="datafile.htm#edges-section">edges section</a> of the datafile
by listing the constraint numbers after the keyword <code>constraint</code>
on the line defining the edge.
Putting an edge
on a constraint does not put its existing vertices on the constraint.
In commands, the status of an edge can be read with the 
"<a href="#on_constraint">on_constraint</a>"
attribute.  The status can be changed with the 
<a href="commands.htm#set">set</a> or <a href="commands.htm#unset">unset</a>
 commands.
Examples:
<pre>
  list edge where on_constraint 2
  set edge constraint 1 where id == 4 or id == 6
  unset edge constraint 3
</pre>

<hr>
<!--==============================Edge on_constraint ============================-->

<a   id="on_constraint,-edge"></a><h2>Edge on_constraint</h2>
See the <a href="#on_constraint" class="keyword">on_constraint</a>
attribute for general elements.
<hr>
<!--============================== Edge e_constraint_list ============================-->

<a   id="e_constraint_list"></a><h2>Edge e_constraint_list</h2>
This read-only attribute gives access to the list of constraints an
edge is on. e_constraint_list[1] is the number of constraints in
the list, followed by the numbers of the constraints.  Note that for
named constraints, the internally assigned numbers are used.

<hr>
<!--============================== Edge boundary ============================-->

<a   id="boundary,-edge"></a><h2>Edge boundary</h2>
<a href="#edges">Edge</a> read-write attribute.
          If an edge is on a 
<a href="constrnt.htm#boundary">parametric boundary</a>,
 then any edges and vertices
          generated from the edge will inherit the boundary.  By default,
          new vertex parameter values are calculated by extrapolating
          from one end of the edge.  This avoids wrap-around problems
          that would arise from interpolating parameter values.  But if
          the interp_bdry_param toggle is on, then interpolation is used.
The status of whether an edge is on a 
boundary can be
queried with the <a href="syntax.htm#boolean-ops">Boolean</a> attribute <a href="#on_boundary">on_boundary</a>.
  Edges can be <a href="commands.htm#unset">unset</a>
from boundaries, and set on them (but care is needed to do this properly).
Examples:
<pre>
  list edges where on_boundary 1
  unset edges boundary 2
</pre>

<hr>
<!--============================= Edge on_boundary =========================-->

<a   id="edge-on_boundary"></a><h2>Edge on_boundary</h2>
See <a href="#on_boundary" class="keyword">on_boundary</a> for general elements.

<hr>
<!--============================== Edge e_boundary ============================-->

<a   id="e_boundary"></a> <h2>Edge e_boundary</h2>
<a href="#edges">Edge</a> read-only integer attribute. 
Internal edge attribute holding the id numbers of the
boundary an edge is on.  Recall that named boundaries
have internal id numbers, which are used here.

<hr>
<!--============================== Edge color ============================-->

<a   id="color,-edge"></a><h2>Edge color</h2>
<a href="#edges">Edge</a> read-write attribute.   
<a href="syntax.htm#colors">Color</a> for graphics.  
The default color is black. Color may be set in the 
<a href="datafile.htm#edges-section">datafile</a>, or with
the <a href="commands.htm#set">set</a> command.
In <a href="graphics.htm#geomview">geomview</a>, the edge color
will show up only for edges satisfying the 
<a href="commands.htm#show">show</a> edge condition, and then
they will have to compete with the edges geomview draws, unless you
turn off geomview's drawing of edges with "ae" in the geomview window.
Examples:
<pre>
  set edge color red where length &gt; 1
  show edge where color != black
</pre>
<hr>
<!--============================== Edge edge_vector ============================-->

<a   id="edge_vector"></a>
<a   id="edge-vector"></a><h2>Edge edge_vector</h2>
<a href="#edges">Edge</a> read-only attribute. The components of the edge vector
in the <a href="model.htm#linear-model">linear model</a> can be accessed
as edge attributes x,y,z or x1,x2,x3,..., or x[1],x[2],x[3].  
edge_vector is another way to refer to the vector as a vector; it is useful
for expressions like dot products where x won't work.
 In a command, the vector between
edge endpoints is used in <a href="model.htm#quadratic-model">quadratic model</a>
or <a href="model.htm#Lagrange-model">lagrange model</a>.  But when used in an
integral, the tangent is evaluated at the Gaussian integration points.
Not defined in the
<a href="model.htm#simplex-model">simplex model</a>. Example to list nearly
vertical edges:
<pre>   list edges where z^2 &gt; 10*(x^2 + y^2)
   print edge[1].edge_vector * edge[2].edge_vector
</pre>
<hr>
<!--============================== Edge no_refine ============================-->

<a   id="no_refine,-edge"></a><h2>Edge no_refine</h2>
<a href="#edges">Edge</a> and <a href="#facets">facet</a>
read-write Boolean attribute.  An edge with the "no_refine" attribute
will not be refined by the <a href="single.htm#r">r</a> command.
This is useful for avoiding needless refining of lines or planes
that are used only for display.  The <code>no_refine</code> attribute
may be specified on the datafile line for an edge, or
the <a href="commands.htm#set" class="keyword">set</a> command may be used.
Examples:
<pre>  set edge no_refine where fixed
  unset edge[2] no_refine
  list edge where no_refine
  print edge[3].no_refine
</pre>

<hr>
<!--============================ edge no_transform =============================-->

<a   id="no_transform,-edge"></a><h2>Edge no_transform</h2>
<a href="#edges">Edge</a> and <a href="#facets">facet</a>
read-write Boolean attribute. An edge or facet with the
"no_transform" attribute will not be duplicated by the view_transform
mechanism; only the original element will occur.  For example, you might
have edges that form a display of coordinate axes, which you would not want
duplicated.
Example:
<pre>   set edge no_transform where valence == 0
</pre>

<hr>
<!--============================== Edge wrap ============================-->

<a   id="wrap"></a><h2>Edge wrap</h2>
<a href="#edges">Edge</a> read-write attribute. When a 
<a href="model.htm#symmetry-groups">symmetry group</a>
is in effect (such as the <a href="model.htm#torus-model">torus model</a>)
 and an edge crosses the boundary of a fundamental domain,
the edge is labelled with the group element that moves the edge head
vertex to its proper position relative to the tail vertex.  The label
is internally encoded as an integer, the encoding peculiar 
to each symmetry group. 
Edge wrappings are set in the <a href="datafile.htm#edges-section">datafile</a>.
The <a href="model.htm#torus-model">torus model</a>
 has its own peculiar wrap representation in the datafile:
<code>*</code> for no wrap, <code>+</code> for positive wrap, and <code>-</code>
for negative wrap.
Wraps are maintained automatically by Evolver during surface manipulations.
The numeric edge wrap values can be queried with attribute syntax. Example:
<pre>
  list edge where wrap != 0
</pre>
Unfortunately, the torus model wraps come out rather opaquely, since
one cannot print hex.  The torus wrap number is the sum of numbers
for the individual directions: +x = 1; -x = 31; +y = 64; -y = 1984;
+z = 4096; -z = 127040.
Caution: even though this attribute can be written by the user at runtime,
only gurus should try it.



<hr>
<!--============================== Edge wrap_list ============================-->

<a   id="wrap_list"></a> <h2>Edge wrap_list</h2>
<a href="#edges">Edge</a> read-write integer array attribute. 
In a torus or symmetry model, this holds the
integer used to encode the wrap of an edge. Note that this is implemented
as an array of length 1 rather than just a value; I forget why, maybe so
it could be length 0 if not needed.
Example:  
<pre>  print edge[3].wrap_list[1] 
</pre>


<!--============================== Edge show ============================-->

<hr>
<a   id="show,-edge"></a><h2>Edge show</h2>
<a href="#edges">Edge</a> and <a href="#facets">facet</a> 
read-only Boolean attribute giving the current status of an
edge or facet according to the <a href="commands.htm#show">show edge</a>
 or <a href="commands.htm#show">show facet</a>
criterion in effect.

<hr>
<!--============================== Edge orientation ============================-->

<a   id="orientation,-edge"></a><h2>Edge orientation</h2>
<a href="#edges">Edge</a> integer read-write attribute. 
              Controls  the sign of oriented integrals on
	      an edge.  Value +1 or -1.  Useful when triangulation
manipulations create an edge going the wrong way.
Example:
<pre>  set edge[2] orientation -1
</pre>


<hr>
<!--============================== Edge frontbody ============================-->

<a   id="frontbody,-edge"></a><h2>Edge frontbody</h2>
<a href="#edges">Edge</a> read-only integer attribute. In the 
<a href="model.htm#string-model">string model</a>, this is the id number
of the body attached to the front of the edge, that is, the body on the 
facet that has positive orientation with respect to the edge.  Invalid
in the <a href="model.htm#soapfilm-model">soapfilm model</a>.

<hr>
<!--============================== Edge backbody ============================-->

<a   id="backbody,-edge"></a><h2>Edge backbody</h2>
<a href="#edges">Edge</a> read-only integer attribute. In the 
<a href="model.htm#string-model">string model</a>, this is the id number
of the body attached to the back of the edge, that is, the body on the 
facet that has negative orientation with respect to the edge.  Invalid
in the <a href="model.htm#soapfilm-model">soapfilm model</a>.

<!--============================== Edge dihedral ============================-->

<a   id="dihedral,-edge"></a><h2>Edge dihedral</h2>
<a href="#edges">Edge</a> read-only real attribute.
The angle in radians between the normals of two facets on an edge. Zero if there
are not exactly two facets.  This attribute is not stored, but
recalculated each time it is used.  If there are not exactly two facets on
the edge, the value is 0.  



<hr>
<!--============================== Edge noncontent ============================-->

<a   id="noncontent,-edge"></a><h2>Edge noncontent</h2>
<a href="#edges">Edge</a> read-write boolean attribute.  When set, indicates
this facet should not be used in volume calculations in the soapfilm model
or facet area calculations in the string model.  Useful, for example,
if you want to have edges be part of a body boundary for display purposes,
but want to use constraint integrands for greater accuracy in volume
calculations.
Example:
<pre>   set edge noncontent where on_constraint 1
</pre>

<!--============================= Edge named quantity =========================-->
<hr>
<a   id="edge-named-quantity"></a><h2>Edge named quantity</h2>
See <a href="#quantity-attribute">named quantities</a> for general elements.

<!--============================= Edge on_quantity =========================-->
<hr>
<a   id="edge-on_quantity"></a><h2>Edge on_quantity</h2>
See <a href="#on_quantity">on_quantity</a> for general elements.


<!--======================= Edge named method instances =========================-->
<hr>
<a   id="edge-named-instances"></a><h2>Edge named method instances</h2>
See <a href="#quantity-attribute">named quantities</a> for general elements.

<!--============================= Edge on_method_instance =========================-->
<hr>
<a   id="edge-on_method_instance"></a><h2>Edge on_method_instance</h2>
See <a href="#on_method_instance">on_method_instance</a> for general elements.


<hr>
<!--============================== Edge e_method_list ============================-->

<a   id="e_method_list"></a> <h2>Edge e_method_list</h2>
<a href="#edges">Edge</a> read-only integer array attribute. 
Internal edge attribute holding the id numbers of the method
instances that this edge contributes to.  Size expands as needed.
Read-only.  Use "<a href="commands.htm#set" class="shortcode">set edge method ... </a>" 
or <a href="commands.htm#unset" class="shortcode">"unset edge method ..</a>"
to change the status of a edge.


<!--============================= Edge extra attributes =========================-->
<hr>
<a   id="edge-extra-attributes"></a><h2>Edge extra attributes</h2>
See <a href="#extra-attributes">extra attributes</a> for general elements.



<!--============================== Facet attributes ============================-->

<hr>
<hr>
<h1>Facet attributes</h1>


<!--============================== Facet id ============================-->

<hr>
<H2>Facet id</h2> See <a href="#id" class="keyword">id</a> for general elements.

<!--============================== Facet oid ============================-->

<hr>
<H2>Facet oid</h2> See <a href="#oid" class="keyword">oid</a> for general elements.


<!--============================== Facet original ============================-->

<hr>
<H2>Facet original</h2> See <a href="#original" class="keyword">original</a> 
for general elements.
<hr>
<!--============================== Facet tension ============================-->

<a   id="density,-facet"></a>
<a   id="tension"></a>
<a   id="tension,-facet"></a><h2>Facet tension or density</h2>
<a href="#facets">Facet</a> read-write attribute.  
Energy per unit
area of facet; surface tension. 
Default 0 in <a href="model.htm#string-model">string model</a>,
 1 in <a href="model.htm#soapfilm-model">soapfilm model</a>.
 May be set in the datafile by adding "<code>tension</code> <em>value</em>"
 to the line defining the facet.  The density is inherited by any
 facets generated by refining.  "Tension" and "density" are synonyms.
Examples:
<pre>
  set facet tension 3 where original == 1
  list facet where density &lt; .4
</pre>

<!--============================== Facet area ============================-->

<hr>
<a   id="area,-facet"></a><h2>Facet area</h2>
<a href="#facets">Facet</a> read-only attribute.
The area of the facet.  Example:
<pre>  list facet where area &lt; .1
</pre>

<hr>
<!--============================== Facet fixed ============================-->

<a   id="fixed,-facet"></a><h2>Facet fixed</h2>
<a href="#facets">Facet</a> read-write attribute.
For a facet to be "fixed" means that any vertex, edge, or facet created
by refining a facet will inherit the fixed attribute.  Fixing a facet
in the datafile or at the command prompt does not fix any edges or
vertices.
A face may be declared fixed in the datafile
by putting <code>fixed</code> on the line defining the face, after the
coordinates.  From the command prompt, one can fix or unfix facets
with the <a href="commands.htm#fix">fix</a> and
<a href="commands.htm#unfix">unfix</a> commands.
<hr>
<!--============================== Facet vertices ============================-->

<a   id="facet-vertices"></a><h2>Facet vertices</h2>
<a href="#facets">Facet</a> read-only attribute. 
<a href="commands.htm#generators">Generates</a>
vertices
around a facet, oriented as the facet boundary. "vertex" and
"vertices" are synonymous.  In the string model, if the facet is not
a closed loop of edges, the vertices will be generated in order
from one end.  If the given facet has negative orientation, then
the vertices will be generated accordingly. Example:
<pre>  list facet[3].vertex
</pre>
<hr>
<!--============================== Facet edges ============================-->

<a   id="facet-edges"></a><h2>Facet edges</h2>
<a href="#facets">Facet</a> read-only attribute. 
<a href="commands.htm#generators">Generates</a>
edges
around a facet, oriented as the facet boundary.  "edge" and "edges"
are synonymous. In the string model, if the edges of the facet do not 
make a closed loop, then the edges will be listed in order starting
from one end.  If the given facet has negative orientation, the edges will
be listed accordingly.  Example:
<pre>  list facet[3].edges
  list facet[-3].edges
</pre>

<hr>
<!--============================== Facet bodies============================-->

<a   id="facet-bodies"></a><h2>Facet bodies</h2>
<a href="#facets">Facet</a> body generator attribute.  
<a href="commands.htm#generators">Generates</a> the bodies
adjacent to a facet, in frontbody-backbody order.
Example:
<pre>
   list facet[2] bodies
</pre>
<hr>
<!--============================== Facet frontbody ============================-->

<a   id="frontbody,-facet"></a><h2>Frontbody</h2>
<a href="#facets">Facet</a> read-write attribute. 
The id of the body of which the facet is on the
		 positively oriented boundary.  Useful
after creating a new body with the <a href="commands.htm#new_body">
new_body</a> command. As a read attribute, the value is 0 if
there is no such body. Examples:
<pre>
  newb := new_body; set facet frontbody newb where color == red
  print facet[2].frontbody
</pre>
Frontbody also works for adding edges to a facet in the string model,
but the added edge must be attach to one end of the edge arc, or
close the arc.

<hr>
<!--============================== Facet backbody ============================-->

<a   id="backbody,-facet"></a><h2>Backbody</h2>
<a href="#facets">Facet</a> read-write attribute. 
The id of the body of which the facet is on the
		 negatively oriented boundary.  Useful
after creating a new body with the <a href="commands.htm#new_body">
new_body</a> command. As a read attribute, the value is 0 if
there is no such body. Examples:
<pre>
  newb := new_body; set facet[1] frontbody newb;
  set facet backbody newb where id == 2 or id == 4;
  print facet[4].backbody
</pre>
Backbody also works for adding edges to a facet in the string model,
but the added edge must be attach to one end of the edge arc, or
close the arc.
<hr>
<!--============================== Facet f_body_list ============================-->

<a   id="f_body_list"></a>
<h2>f_body_list</h2>
<a href="#facets">Facet</a> internal array attribute. 
Contains the frontbody and backbody ids of the facet.  Not too useful directly;
listed here because it shows up in <a href="commands.htm#list-attributes"
class="keyword">list_attributes</a>.


<hr>
<!--============================== Facet valence ============================-->

<a   id="valence,-facet"></a><h2>Facet valence</h2>
<a href="#facets">Facet</a> read-only attribute.
The valence of a facet is the number of edges (or vertices)
that it contains.  Most useful in the 
<a href="model.htm#string-model">string model</a>.  Example:
<pre>
  list facets where valence != 3
</pre>
<hr>
<!--============================== Facet constraints ============================-->

<a   id="constraints,-facet"></a><h2>Facet constraints</h2>
<p>
<a href="#facets">Facet</a> read-write attribute.
Putting a facet on a 
<a href="constrnt.htm#level-set-constraints">constraint</a>
 means that every vertex, edge, or facet
generated by refining the facet will inherit that constraint.  Setting
a facet on a constraint does not set any of its existing edges or vertices
on the constraint.  Facets may be put on constraints in the 
<a href="datafile.htm#faces-section">datafile</a> by listing the 
constraint numbers after the keyword <code>constraint</code> on the line
defining the facet, or with the
<a href="commands.htm#set">set</a> command.  They may be removed
with the <a href="commands.htm#unset">unset</a> command.
Examples:
<pre>
  list facets where on_constraint 1
  set facet[2] constraint 2
  unset facet constraint 1
</pre>
<hr>
<!--============================== Facet on_constraint ============================-->

<a   id="facet-on_constraint"></a><h2>Facet on_constraint</h2>
See the <a href="#on_constraint" class="keyword">on_constraint</a>
attribute for general elements.
<hr>
<!--============================== Facet f_constraint_list ============================-->

<a   id="f_constraint_list"></a><h2>f_constraint_list</h2>
This read-only attribute gives access to the list of constraints a
facet is on. f_constraint_list[1] is the number of constraints in
the list, followed by the numbers of the constraints.  Note that for
named constraints, the internally assigned numbers are used.


<hr>
<!--============================== Facet boundary ============================-->

<a   id="boundary,-facet"></a><h2>Facet boundary</h2>
<a href="#facets">Facet</a> read-write attribute.
          If a facet is on a 
<a href="constrnt.htm#boundary">parametric boundary</a>,
 then any facets, edges, and vertices
          generated from the facet will inherit the boundary. By default,
          new vertex parameter values are calculated by extrapolating
          from one vertex of the facet.  This avoids wrap-around problems
          that would arise from interpolating parameter values.  But if
          the interp_bdry_param toggle is on, then interpolation is used.
The status of whether a facet is on a 
boundary can be
queried with the <a href="syntax.htm#boolean-ops">Boolean</a> attribute 
<a href="#on_boundary">on_boundary</a>.  The actual boundary number is stored in
the attribute f_boundary, which can be read but should not be written
directly.
  Facets can be <a href="commands.htm#unset">unset</a>
from boundaries, and set on them (but care is needed to do this properly).
Examples:
<pre>
  list facets where on_boundary 1
  unset facet[2] boundary 2
</pre>

<hr>
<!--============================== Facet f_boundary ============================-->

<a   id="f_boundary"></a><h2>Facet f_boundary</h2>
<a href="#facets">Facet</a> read-only integer attribute.  The number of any 
<a href="constrnt.htm#boundary">parameterized boundary</a>
 the facet is on.  Note that named boundaries have internal
numbers, and those are used here.


<hr>
<!--============================== Facet color ============================-->

<a   id="color,-facet"></a><h2>Facet color</h2>
<a href="#facets">Facet</a> read-write attribute.   
<a href="syntax.htm#colors">Color</a> of both sides of facet for graphics.
Default is white.
Datafile example:
<pre>  Faces
  1   1 2 3 color red
</pre>
Command examples:
<pre>  list facets where color == red
  set facet[3] color green
  set facet color red where area &gt; 2
</pre>
<hr>
<!--============================== Facet frontcolor ============================-->

<a   id="frontcolor"></a><h2>Facet frontcolor</h2>
<a href="#facets">Facet</a> read-write attribute.   
<a href="syntax.htm#colors">Color</a> of positive side of facet for graphics.
Default is white.
Datafile example:
<pre>  Faces
  1   1 2 3 frontcolor green backcolor red
</pre>
Command examples:
<pre>  list facets where frontcolor == red
  set facet[3] frontcolor green
  set facet frontcolor red where area &gt; 2
</pre>
<hr>
<!--============================== Facet backcolor ============================-->

<a   id="backcolor"></a><h2>Facet backcolor</h2>
<a href="#facets">Facet</a> read-write attribute.   
<a href="syntax.htm#colors">Color</a> of negative side of facet for graphics.
Default is white.  Set also when the "color" attribute is set.
Datafile example:
<pre>  Faces
  1   1 2 3 frontcolor green backcolor red
</pre>
Command examples:
<pre>  list facets where backcolor == red
  set facet[3] backcolor green
  set facet backcolor red where area &gt; 2
</pre>

<hr>
<!--============================== Facet opacity ============================-->

<a   id="opacity"></a><h2>Facet opacity</h2>
<a href="#facets">Facet</a> read-write attribute for transparency. Syntax:
   set facet opacity value where condition
where value is between 0 (clear) and 1 (opaque).  Screen graphics will
show transparency, but PostScript output will not.   Hitting the 'O'
key in the graphics window will toggle transparency, if the opacity
attribute has been assigned values.
Datafile example:
<pre>  Faces
  1   1 2 3  opacity 0.5
</pre>
Command examples:
<pre> 
  set facet opacity 0.6
  set facet opacity 0.6 where original == 2
</pre>

<hr>
<!--============================== Facet facet_normal ============================-->

<a   id="facet_normal"></a>
<a   id="facet-normal"></a><h2>Facet facet_normal</h2>
<a href="#facets">Facet</a> read-only attribute. 
The components of the facet normal vector may be referred to as
x,y,z or x1,x2,x3 or x[1],x[2],x[3]
in the <a href="model.htm#linear-model">linear model</a>.  Length is equal
to facet area.  facet_normal is the internal name of the attribute,
and is useful to refer to the entire vector in array expressions
such as dot product.
In <a href="model.htm#quadratic-model">quadratic model</a>
or <a href="model.htm#Lagrange-model">lagrange model</a>, only the three facet
corner vertices are used to calculate the normal.  When used in 
integrals, the normal is calculated at each integration points.
  Not defined in 
<a href="model.htm#simplex-model">simplex model</a>. 






<hr>
<!--============================== Facet no_display ============================-->

<a   id="no_display"></a>
<a   id="nodisplay"></a><h2>Facet no_display</h2>
<a href="#facets">Facet</a> read-write attribute.   
When set, suppresses the display of the facet in graphics.  Can
be set in the <a href="datafile.htm#faces-section">datafile</a>
by adding <code>nodisplay</code> to the line defining the facet. Can
also be manipulated by the <a href="commands.htm#set">set</a>
and <a href="commands.htm#unset">unset</a> commands. <code>No_display</code>
is a synonym provided since that's what I kept typing in.
 Example:
<pre>   set facet nodisplay where color != red
</pre>
<hr>
<!--============================== Facet no_refine ============================-->

<a   id="no_refine,-facet"></a><h2>Facet no_refine</h2>
<a href="#facets">Facet</a>
read-write Boolean attribute.  Giving a facet the <code>no_refine</code>
attribute has no effect except that edges created within the
facet by refining will inherit the <code>no_refine</code> attribute.  So to
avoid refinement of a plane, all edges and facets in the plane
must be given the <code>no_refine</code> attribute.  The <code>no_refine</code> attribute
may be specified on the datafile line for a facet, or
the <a href="commands.htm#set" class="keyword">set</a> command may be used.
Examples:
<pre>  set facet no_refine where fixed
  unset facet[2] no_refine
  list facet where no_refine
  print facet[3].no_refine
</pre>

<hr>
<!--============================ facet no_transform =============================-->

<a   id="no_transform,-facet"></a><h2>Facet no_transform</h2>
<a href="#edges">Edge</a> and <a href="#facets">facet</a>
read-write Boolean attribute. An edge or facet with the
"no_transform" attribute will not be duplicated by the view_transform
mechanism; only the original element will occur.  For example, you might
have facets that form a display of outer walls which you would not want
duplicated.
Example:
<pre>   set facet no_transform where color == brown
</pre>

<hr>
<!--============================== Facet orientation ============================-->

<a   id="orientation,-facet"></a><h2>Facet orientation</h2>
<a href="#facets">Facet</a> read-write attribute. 
              Controls  the sign of oriented integrals on
	      a facet.  Value +1 or -1.  Useful when triangulation
manipulations create a facet with an undesired orientation.
Example:
<pre>   set facet[123] orientation -1
</pre>
Also see the <a href="commands.htm#reverse_orientation" class="keyword">
reverse_orientation</a> command to physically reverse a facet's orientation.

<hr>
<!--============================== Facet noncontent ============================-->

<a   id="noncontent,-facet"></a>
<a   id="facet-noncontent"></a><h2>Facet noncontent</h2>
<a href="#facets">Facet</a> read-write attribute.  When set, indicates
this facet should not be used in volume calculations.  Useful, for example,
if you want to have facets be part of a body boundary for display purposes,
but want to use constraint integrands for greater accuracy in volume
calculations.
Example:
<pre>   set facet noncontent where on_constraint 1
</pre>

<hr>
<!--============================== Facet phase ============================-->

<a   id="phase,-facet"></a><h2>Facet phase</h2>
<a href="#facets">Facet</a> read-write attribute. 
     If there is a <a href="datafile.htm#phase-decl">phasefile</a>,
      this attribute determines the
edge tension of an edge between two facets in the string model.
Example:
<pre>   list facet where phase == 1
</pre>

<!--============================= Facet named quantity =========================-->
<hr>
<a   id="named-quantity,-facet"></a><h2>Facet named quantity</h2>
See <a href="#quantity-attribute">named quantities</a> for general elements.

<!--============================= Facet on_quantity =========================-->
<hr>
<a   id="on_quantity-facet"></a><h2>Facet on_quantity</h2>
See <a href="#on_quantity">on_quantity</a> for general elements.


<!--======================= Facet named method instances =========================-->
<hr>
<a   id="named-instances,-facet"></a><h2>Facet named method instances</h2>
See <a href="#quantity-attribute">named quantities</a> for general elements.

<!--============================= Facet on_method_instance =========================-->
<hr>
<a   id="facet-on_method_instance"></a><h2>Facet on_method_instance</h2>
See <a href="#on_method_instance">on_method_instance</a> for general elements.



<!--============================= Facet extra attributes =========================-->
<hr>
<a   id="extra-attributes,-facet"></a><h2>Facet extra attributes</h2>
See <a href="#extra-attributes">extra attributes</a> for general elements.

<!--============================== Facet f_method_list ============================-->

<hr> <a   id="f_method_list"></a><h2>Facet f_method_list</h2>
<a href="#facets">Facet</a> array attribute.  Internal name
for the array holding the id numbers of the method instances
to which this facet contributes.  Read-only.

<hr>
<!--============================== Facet f_next_bfacet ============================-->

<a   id="f_next_bfacet"></a><h2>Facet f_next_bfacet</h2>
<a href="#facets">Facet</a> read-only attribute.  Used internally
when iterating over the facets comprising a body.
 

<!--============================== Facet f_next_vfacet ============================-->

<hr> <a   id="f_next_vfacet"></a><h2>Facet f_next_vfacet</h2>
<a href="#vertices">Vertex</a> integer attribute.  Internal attribute
used for the linked list that is used when iterating over the facets
adjacent to a vertex.  Not accessible to user since it uses internal 
element id type.




<!--============================== Body attributes ============================-->


<hr>
<hr>
<h1>Body attributes</h1>
<!--============================== Body id ============================-->


<hr>
<H2>Body id</h2> See <a href="#id" class="keyword">id</a> for general elements.

<!--============================== Body original ============================-->

<hr>
<H2>Body original</h2> See <a href="#original" class="keyword">original</a> 
for general elements.


<hr>
<!--============================== Body facets ============================-->


<a   id="body-facets"></a><h2>Body facets</h2>
<a href="#bodies">Body</a> read-only attribute. 
<a href="commands.htm#generators">Generates</a>
facets
bounding a body, with proper facet orientation with respect to the body. 
Example:
<pre>  list body[1].facets
</pre>
<hr>
<!--============================== Body density ============================-->

<a   id="density,-body"></a><h2>Body density</h2>
<a href="#bodies">Body</a> read-write attribute. 
              Density used for gravitational potential energy.
It can be set in the <a href="datafile.htm#bodies-section">bodies section</a>
of the datafile, or with the <a href="commands.htm#set">set</a> command,
or by assignment.  Command examples:
<pre>  print body[2].density
  set body density 3
  body[2].density := 5
</pre>

<hr>
<!--============================== Body volume ============================-->

<a   id="volume"></a>
<a   id="body-volume"></a><h2>Body volume</h2>
<a href="#bodies">Body</a> read-only attribute. 
               Actual volume of a body.  This is the sum of three
parts, in the soapfilm model:
<ul>
<li> An integral over the facets bounding the body. This is 
\int z dx dy normally, but \int (x dy dz + y dz dx + z dx dy)/3
if <a href="datafile.htm#symmetric_content" class="keyword">symmetric_content</a>
 is in effect.
<li> Any constraint content edge integrals applying to the body.
<li> The body's volconst attribute.
</ul>
In the string model, the parts are
<ul>
<li> An integral over the edges bounding the body's facet. This is 
\int -y dx.
<li> Any constraint content vertex integrals applying to the body.
<li> The body's volconst attribute.
</ul>
Body volumes can be displayed with the <a href="single.htm#v">v</a>
command, or with standard attribute syntax. Example:
<pre>  print body[1].volume
  foreach body where volume &gt; 2 do print id
</pre>


<hr>
<!--============================== Body target ============================-->

<a   id="target"></a>
<a   id="target-volume"></a><h2>Body target</h2>
<a href="#bodies">Body</a> read-write attribute. 
           The target volume of a volume constraint.  May be set in 
the 
<a href="datafile.htm#bodies-section">datafile</a>,
 by the <a href="single.htm#b">b</a> command, or the 
<a href="commands.htm#set">set</a> command.
A volume constraint may be removed by the 
<a href="commands.htm#unset">unset</a>, or with the 
<a href="single.htm#b">b</a> command.
Command examples:
<pre>  set body[1] target 23
  unset body target where id == 2
  print body[2].target
</pre>
<hr>

<!--============================== Body volfixed ============================-->

<a   id="volfixed"></a><h2>Body volfixed</h2>
<a href="#bodies">Body</a> Boolean read-only attribute. 
Value is 1 if the volume of the body is fixed, 0 if not.

<hr>
<!--============================== Body volconst ============================-->

<a   id="volconst"></a>
<a   id="body-volconst"></a><h2>Body volconst</h2>
<a href="#bodies">Body</a> read-write attribute. 
     A constant added to the calculated volume. Useful for 
correcting for omitted parts of body boundaries.  Also used 
internally as a correction in the <a href="model.htm#torus-model">torus model
</a>, which will use the target volume to calculate volconst internally.  
In the torus model, the target volume should be set within 1/12 of a
torus volume of the actual volume for each body, so the correct volconst
can be computed.  Each volconst will be adjusted proportionately when
the volume of a fundamental torus domain is change by changing the
period formulas.
Volconst can be set
 in the datafile <a href="datafile.htm#bodies-section">bodies section</a>,
or interactively by the <a href="commands.htm#set">set</a> command or
by assignment.  Examples:
<pre>  print body[1].volconst
  set body[2] volconst 1.2
  body[2].volconst := 1.2
</pre>
It is best to avoid using volconst except in the torus model.  Rather,
use <a href="datafile.htm#constraint-decl">edge content integrals</a>
so that the proper adjustments will be made if the boundary
of the surface is moved, or <a href="commands.htm#rebody">rebody</a>
is done.


<hr>
<!--============================== Body actual_volume ============================-->

<a   id="actual_volume"></a><h2>Body actual_volume</h2>
<a href="#bodies">Body</a> datafile attribute. 
 Actual_volume is a number that can be specified in the
 datafile definition of a body
 in the rare circumstances where the  torus model volume
  volconst calculation gives the wrong answer; volconst
  will be adjusted to give this volume of the body.

<hr>
<!--============================== Body pressure ============================-->

<a   id="pressure,-body"></a><h2>Body pressure</h2>
<a href="#bodies">Body</a> read-write real attribute. 
        If a body has a prescribed volume, this is a read-only attribute,
which is the Lagrange multiplier for the volume constraint.
If a body is given a prescribed pressure, then there is an energy
term equal to pressure times volume.  A body cannot have a prescribed
volume and a prescribed pressure at the same time.  Prescribed 
volume or pressure
can be set in the <a href="datafile.htm#bodies-section">bodies section</a>
of the datafile.  If pressure is prescribed, then the value can be
changed interactively with the <a href="single.htm#b">b</a> command,
the  <a href="commands.htm#set">set</a> command, or by assignment.
Examples:
<pre>  print body[2].pressure
  body[2].pressure := 1.3
  set body[2] pressure 1.3
</pre>


<hr>
<!--============================== Body phase ============================-->

<a   id="phase,-body"></a><h2>Body phase</h2>
<a href="#bodies">Body</a> read-write attribute. 
     If there is a <a href="datafile.htm#phase-decl">phasefile</a>,
      this attribute determines the
facet tension of an edge between two bodies in the soapfilm model.
Example:
<pre>   list body where phase == 1
</pre>


<hr>
<!--============================== Body centerofmass ============================-->

<a   id="centerofmass"></a><h2>Body centerofmass</h2>
<a href="#bodies">Body</a> read-write boolean attribute. 
Boolean body attribute.  Applies to the "connected" bodies mode of graphical
display in the torus model. When this is set for a body, the center of mass
of the body as displayed is remembered, and the next time a body is graphed,
its wrap is such that its new center of mass is near its previous center
of mass.  This prevents bodies near the boundaries of the fundamental
region from jumping back and forth as they shift slightly during evolution.
Default on.  Example:
<pre> set bodies centerofmass </pre>

<!--============================== Body quantity ============================-->
<a   id="named-quantity,body"></a>
<a   id="method-instance,body"></a>
<a   id="on_quantity,body"></a>
<a   id="on_method_instance,body"></a>
<hr> <a   id="b_method_list"></a><h2>Body quantities</h2>
There are no named methods currently implemented for bodies, so named quantities
and methods do not apply.

<!--============================== Facetedge attributes ============================-->

<hr>
<hr>
<h1>Facetedge attributes</h1>

<!--============================== Facetedge id ============================-->

<hr>
<H2>Facetedge id</h2> See <a href="#id" class="keyword">id</a> for general elements.

<!--============================== Facetedge oid ============================-->

<hr>
<H2>Facetedge oid</h2> See <a href="#oid" class="keyword">oid</a> for general elements.

<!--============================== Facetedge edge ============================-->

<a   id="facetedge-edge"></a><h2>Facetedge edge</h2>
<a href="#facetedges">Facetedge</a> read-only attribute. 
<a href="commands.htm#generators">Generates</a> the single  edge
of the facetedge.
Example:
<pre>  print facetedge[1].edge[1].id
</pre>
<hr>

<!--============================== Facetedge facet ============================-->

<a   id="facetedge-facet"></a><h2>Facetedge facet</h2>
<a href="#facetedges">Facetedge</a> read-only attribute. 
<a href="commands.htm#generators">Generates</a> the single  facet
of the facetedge.
Example:
<pre>  print facetedge[1].facet[1].id
</pre>
<hr>


<!--============================== Facetedge nextedge ============================-->


<a   id="nextedge"></a><h2>Facetedge nextedge</h2>
<a href="#facetedges">Facetedge</a> internal attribute. 
Oriented id number of the next facetedge in the facet edge loop.  May be 0 in
the string model for the last edge of a facet. Not available
to users except in the output of a "list facetedges ... " command.
<hr>

<!--============================== Facetedge prevedge ============================-->

<a   id="prevedge"></a><h2>Facetedge prevedge</h2>
<a href="#facetedges">Facetedge</a> internal attribute. 
Oriented id number of the previous facetedge in the facet edge loop.  May be 0 in
the string model for the first edge of a facet. Not available
to users except in the output of a "list facetedges ... " command.
<hr>

<!--============================== Facetedge nextfacet ============================-->

<a   id="nextfacet"></a><h2>Facetedge nextfacet</h2>
<a href="#facetedges">Facetedge</a> internal attribute. 
Oriented id number of the next facetedge in the loop of facets around the edge.  Not available
to users except in the output of a "list facetedges ... " command.
<hr>

<!--============================== Facetedge prevfacet ============================-->

<a   id="prevfacet"></a><h2>Facetedge prevfacet</h2>
<a href="#facetedges">Facetedge</a> internal attribute. 
Oriented id number of the previous facetedge in the loop of facets around an edge.  Not available
to users except in the output of a "list facetedges ... " command.
<hr>


<hr>


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